Opis: A Comparison of Different Line, Geometric Representations for Functions of a Complex Variable a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science, in Candidacy for the Degree of Doctor of Philosophy, (Department or Mathematics) - Gibbens Gladys E. C.

Excerpt from A Comparison of Different Line, Geometric Representations for Functions of a Complex Variable a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science, in Candidacy for the Degree of Doctor of Philosophy, (Department or Mathematics)

Wilczynski has recently given two methods for constructing a congruence of lines determined by a functional relation between two complex variables, which enable one to visualize the properties of the function by studying the properties of the resulting congruence. In the first method, the two complex variables are represented upon two distinct planes, parallel to each other and a unit apart, the corresponding coordinate axes for the two planes being chosen parallel to each other in such a way that the two origins lie upon a line perpendicular to the two planes. If the points of the first plane are joined to the points of the second plane which correspond to them by means of a given functional relation w = F(z), a two-parameter family, or a congruence, of straight lines is obtained. These congruences have certain characteristic properties which hold for the totality of analytic functions, and in addition, of course, special properties which depend upon the choice of the particular function F(z). The developables and focal sheets of such congruences are always imaginary, except in a trivial special case, but some interesting real surfaces are closely associated with them.

The second method of representation makes use of a Riemann sphere. The two complex variables are projected upon the same sphere, and points of the sphere corresponding to each other by means of the function w = F(z) are joined by lines. The congruences obtained in this way always have real focal sheets and developables, and are therefore more interesting than those obtained by the first method.

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Szczegóły: A Comparison of Different Line, Geometric Representations for Functions of a Complex Variable a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science, in Candidacy for the Degree of Doctor of Philosophy, (Department or Mathematics) - Gibbens Gladys E. C.

Nazwa: A Comparison of Different Line, Geometric Representations for Functions of a Complex Variable a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science, in Candidacy for the Degree of Doctor of Philosophy, (Department or Mathematics)
Autor: Gibbens Gladys E. C.
Wydawnictwo: FB &c Ltd
Kod paskowy: 9781330190555
Języki: angielski
Ilość stron: 28
Format: 15.2x22.9cm
Oprawa: Miękka

Recenzje: A Comparison of Different Line, Geometric Representations for Functions of a Complex Variable a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science, in Candidacy for the Degree of Doctor of Philosophy, (Department or Mathematics) - Gibbens Gladys E. C.